Wait, how can air resistance increase acceleration when going up. From what I have understood, air resistance slows down the object flying upward, likewise with gravity.

I do understand that I have to break up the question into the upward and downward parts, but the question asks for the speed just before it hits the ground. Furthermore, the object being shot with an initial velocity will eventually go to rest at its highest point, then fall back down again. This is what caused me to believe that the upward component is negligible.

Without air resistance, the only force is gravity (weight), pointing down.
The acceleration vector, therefore, points downwards.

When the ball goes up, the force of the air resistance points down, adding to the force of gravity and so increasing the acceleration.

When the ball goes down, the air resistance points up, subtracting from the force of gravity and so reducing the acceleration.

When the ball falls from it's highest point to the ground, what determines the speed when it strikes?
Perhaps the ball has enough time to reach it's terminal velocity? How would you know?
Don't you need to know how high it started from?